# FRM Fun 11. Do we really need all three durations? let's settle this now!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
FRM Fun 11.

The Macaulay duration is the weighted average maturity of a bond, where the weights are the present values (as a percentage of the bond's price) of the cash flow. Hull's Table 4.6 below illustrates this nicely; the Macaulay duration of his 3-year bond is 2.653 years, which the sum of the right-hand column values, each of which is the product of a PV cash flow and its weight.

Questions (please do not conduct a forum search, as these questions summarize years of forum threads!):
1. As Macaulay Duration is the bond's weighted average maturity, we comfortably express Mac duration in years; e.g., Macaulay duration of 2.653 years. So much that, upon reading (for example) "bond duration of 7.0 years" the connotation is Macaulay.
But what are the UNITS of modified duration and why?
2. What is the difference between Macaulay duration, modified duration and effective duration; especially effective duration? (Best answers are brief but mathematically precise and decisive)

#### Aleksander Hansen

##### Well-Known Member
ModDur = represents the change in P(y), in percentage terms, where P is the price function and y can be either an endogenous or exogenous variable. The changes to the pricing function P(y) is, typically given in terms of a % basis point change to the an exogenous variable y, or alternatively as a simple percentage change in one basis point (common in trading). The units differ since the MacDur is just the weighted average maturity, but ModDur measures the sensitivity wrt the derivative y.

In the special case where we compound continuously we can see that for ModDur = - [1/P x dP/dy] we get the result:
k=1, n, Σ(t_k * e^(-yt_k)"CF"_k)/P = DurMacaulay.
[yes, it's ugly but don't know how to upload or do math equations here from an iPad. Type it into Wolfram Alpha and it should look better]
Looking at it from the discrete case we know that ModDur = MacDur/(1 + [y/k]), and as lim k --> infinity, the term y/k tends to zero. The derivation is similar (but longer so I'll skip it) to the continuous case, and it should be seen that it follows immediately.

Effective duration is somewhat unprecise so is not much used in practice, however, the idea is to approximate some discrete P(y), such that we look at the change in P(y) when we have a small down shift in yields, as well as a small up shift in yields, that is 1/2* {P(-dy) - P(+dy)}/[P0*dy], where dy is the new yield after a small shift. (We write P(-dy)... ince price is monotonically decreasing in yield.)

This would be appropriate to use (although superior techniques exist) when our pricing function violates the smooth-pasting condition, and the function is strictly less than C^2 differentiable. Typical example would be negative convexity.

****************
Addendum to question: what is the equivalent of negative convexity?
Are all functions that are not convex, concave? Would a semi- and hemi-convex function always be convex? Can a function be both semi-convex and semi-concave? Which condition is stronger: semi convex or convex?

#### ShaktiRathore

##### Well-Known Member
Subscriber
Macaulay Duration is basically number of years it would take for the investment in the fixed income security to be recovered.It is max. for zero coupon bond since it takes the max. years equal to maturity of bond for the invested money to be recovered. Bonds with large initial outlay of money have lower bond macaulay durations.
Mathematically, Bond Price=P=[C/(1+y)+C/(1+y)^2+.........+F+C/(1+y)^n]
differentiating w.r.t y ; dP/dy=-(1/1+y)*(C/(1+y)+2*C/(1+y)^2+.........+n*F+C/(1+y)^n)
multiplying both sides by 1/P
1/P*dP/dy=-(1/1+y)*1/P*(C/(1+y)+2*C/(1+y)^2+.........+n*F+C/(1+y)^n)
or (dP/P)/dy=-(1/1+y)*Macaulay Duration
Now (dP/P)/dy is nothing but the modified duration which is percentage change in price of bond with percent change in yield and y is the periodic yield for the bond. Modified duration is expressed in yrs.
Effective Duration is used to analyse the average percent change in bond prices with average change of +-change in yield. for y-x let price be Bond Price calculated be P1. For another shift in yield to y+x the Bond Price found be P2.
Hence effective yield=Average percent change in price/Average change in yield
= ((P2-P)/P+(P-P1)/P)/2/(2*x/2)
=P2-P1/2*x*P

#### Aleksander Hansen

##### Well-Known Member
dP/dy=-(1/1+y)*Macaulay Duration
Now dP/dy is nothing but the modified duration which is percentage change in price of bond with percent change in yield and y is the periodic yield for the bond. Modified duration is expressed in percent.

dP/dy=-(1/1+y)*Macaulay Duration?
Now dP/dy is nothing but the modified duration

The two statement seem to be mutually exclusive.
For the former I would write dP/dy= - P(y)*MacDur/(1+y/k)
and the latter seem to be missing the -1/P?

#### ShaktiRathore

##### Well-Known Member
Subscriber
dP/dy=-(1/1+y)*Macaulay Duration?
Now dP/dy is nothing but the modified duration

The two statement seem to be mutually exclusive.
For the former I would write dP/dy= - P(y)*MacDur/(1+y/k)
and the latter seem to be missing the -1/P?
Yeah Aleksander,
I forgot to take 1/P on both sides of equation and thanks for letting me know this.I have edited it.

##### New Member
As above posts adequately explained in mathematical terms.. would like to just add in more 'qualitative' terms..

The type of duration measure used is dependent upon the type of investments being analyzed (e.g., bullet securities versus bonds with embedded options) among many others.and hence all three will be interpreted differently within the context of the security being analyzed.

Macaulay duration has temporal characteristics i.e. it has units of the period (years). Modified duration is a more meaningful compared to macaulay duration as it 'truely' provides us a measure of the responsiveness(sensitivity) of a bond’s price to interest rate changes. As already explained above mathematically, both have a closed form solutions for a given price function. Here point to keep in mind is that the solution assumes no change in bond's cash flow due to interest rate changes.

Both MacDuration and Mod. duration can be used only in case of bonds which have certainty in their cash flows. Therfore for a coupon paying bond, these durations can be interpreted more meaningfully. However, in case of bond with uncertain cash flows, such as callable bond, modified duration analysis will be flawed due to an embedded option. Here, effective duration is more meaningful as effective duration takes into account not only complex stochastic variations in interest rate term structure model but also change in value of embedded options due to interest rate change.

#### Aleksander Hansen

##### Well-Known Member
I would add that nowadays when you speak of effective duration you have to be careful; it is often taken to mean average effective duration, which is also known as option adjusted duration which incorporates convexity, and more specifically prepayments, coupons that can reset/cash flow changes as well as puts.

#### Aleksander Hansen

##### Well-Known Member
Just to address the basic question without the technicalities above: modified duration is measured in years.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Thanks for great replies here!
1. For several years I referred to modified duration as unitless, as this is what i was taught. But last year pitzetch corrected me and showed me why the units of modified duration are also expressed in years: http://forum.bionicturtle.com/threads/l2-t5-25-duration.3447/
2. In regard to effective duration, I agree with Aleks that it has a loose definition. But in the FRM, assigned Veronesi is at least consistent with Fabozzi in the sense that effective duration is required when the bond's cash flows are variant to the interest rate, so that unlike a vanilla bond (where rate impacts price via discounting), in an MBS/mortgage (for example), the interest rate (yield, in most cases) is impacting both via discounting and via discounted cash flows. Consequently, this implies that mac duration cannot be access analytically as the function of the first derivative dP/dy. So, effective duration, the way that i think about it, computes the secant line that approximates the tangent line (which itself cannot be accessed analytically). Effective duration re-prices (a "simulation") the bond at two nearby yields and then simply finds the slope of the line between them (the secant) such that: = slope *-1/P = rise/run*-1/P = (change in price)/(change in yield) ~= dP/dy*-1/P.

or, put another way: if we can't get, or are not justified in retrieving, modified duration by dP/dy*1/P, we resort to approximating (we approximate the linear approximation!) by retrieving the slope of the nearby secant.

In this way, effective duration may be a necessary approximation to modified duration, but ultimately has the same meaning (and it's imprecision is tolerable as modified duration is a linear approximation in the first place.)
But thank you Aleks for the more robust condition, i learned something here:
This would be appropriate to use (although superior techniques exist) when our pricing function violates the smooth-pasting condition, and the function is strictly less than C^2 differentiable [ <-- yes of course!] Typical example would be negative convexity.

Finally, I hope candidates noticed above that under the case of continuous compounding, Mac = modified duration. Note this follows from Mod duration = mac duration/(1+y/k) where k = periods per year but k --> infinite under continuous compounding.

#### Mark W

##### Active Member
Hi guys,

First BT forum post...and it's just a little clarification question.

We have: Modified Duration = (dP/P)/dy = -(1/1+y)*Macaulay Duration as above

We don't have a minus sign?

Thanks

Edit: P.S. David, is it possible to mark whether questions relate to P1 or P2...?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi maw,

They are the same. dP/dy = dollar duration = slope of the tangent and is negative. In Mod Duration = -1/P*dP/dy, the leading negative converts a negative dP/dy into positive years.
• your reference to my 2007 article uses Fabozzi's effective duration which, in computing the slope of the tangent, uses P(-yield shock) - P(+yield shock) in the numerator. As P(-yield shock) is greater, this numerator is already positive.
• The above, consistent with currently assigned Veronesi, gives the same answer but is also (IMO) slightly superior for its rigor: Mod Duration = -1/P*slope. The slope (dollar duration) in fact is negative, so it's a more logical to negate the negative into a positive (rather than solve for the positive directly). Although it's the same in the end. I hope that helps, thanks,

#### Mark W

##### Active Member
Perfect, thanks David.

Mark

#### bhar

##### Active Member
Hi

To summarise - Macaulay Duration is Time wieghted duration. Effective Duration is the duration calculated as a sensitivity measure with respect to movement in interest rates. However, I am not able to see a clear definition of Modified Duration in this discussion. Could anyone please explain. Thanks in advance.

2) ' Note this follows from Mod duration = mac duration/(1+y/k) where k = periods per year but k --> infinite under continuous compounding.'

This is a open ended item. how can we divide the y with k ?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi bhar,
1. Both effective and modified duration are the (linear) sensitivity of bond price to change in yield; their interpretation is the same (both mathematically aspire to the same dP/dy*1/P).

The difference is the method: modified duration can find this sensitivity analytically (directly), dP/dY*1/P, with Mac Duration/(1+y/k).

Effective duration requires an approximation, using a full re-pricing of the bond given the yield shock, in the case where bonds (eg, MBS) produce future cash flows that vary with yields.

We might view mod duration as the function of the precise tangent line, but effective duration as the more general (robust) approximation of the same (a function of the very nearby secant that approximates the tangent)
2. y is yield, k = compound periods per year. For example, if yield is 6%, and semi-annual coupons, then k = 2, and mod duration = mac duration/(1+6%/2).

#### bhar

##### Active Member
However, since in mod duration we are viewing k as a infinite or continuous. How do we solve this mathematically.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
bhar - if compound freq is continuous (is not required), then k is infinite (periods per year), and then Mod duration = Mac Duration/(1+y/k) reduces to:
Mod duration = Mac Duration/(1+y/inf), but y/inf tends --> 0, so Mod duration = Mac Duration/(1+0) = Mac Duration. In the special case of continuous compounding, mod duration = mac duration, but for all discrete compound frequencies (e..g., semi-annual), Mod duration < Mac duration

#### bhar

##### Active Member
Thanks. That settles my understanding.

#### tallerico

##### New Member
Thanks for great replies here!
1. For several years I referred to modified duration as unitless, as this is what i was taught. But last year pitzetch corrected me and showed me why the units of modified duration are also expressed in years: http://forum.bionicturtle.com/threads/l2-t5-25-duration.3447/
can someone explain how i can g et on the link above with regard to modified duration units so i can read it.
Thanks Tallerico

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
tallerico -- links to the Practice Question Boards (which includes the link to which you refer) are not free to the public: paid members (of any FRM product) have seamless access due to their account status. The upper boards, like this one, are free to public. Thanks!

#### Andrew

##### New Member
Hi Everyone! What is the intuition behind modified duration being Macaulay duration divided by (1+Yield/K)?